Searching components with simple structure in simultaneous component analysis: Blockwise Simplimax rotation

Abstract Simultaneous component analysis (SCA) is a fruitful approach to disclose the structure underlying data stemming from multiple sources on the same objects. This kind of data can be organized in blocks. To identify which component relates to all, and which to some sources, the block structure in the data should be taken into account. In this paper, we propose a new rotation criterion, Blockwise Simplimax, that aims at block simplicity of the loadings, implying that for some components all variables in a block have a zero loading. We also present an associated model selection criterion, to aid in selecting the required degree of simplicity for the data at hand. An extensive simulation study is conducted to evaluate the performance of Blockwise Simplimax and the associated model selection criterion, and to compare it with a sparse competitor, namely Sparse group SCA. In the conditions considered Blockwise Simplimax performed reasonably well, and either performed equally well as, or clearly outperformed Sparse group SCA. The model selection criterion performed well in simple conditions. The usefulness of Blockwise Simplimax and Sparse group SCA is illustrated using sensory profiling data regarding different cheeses.